Pure mathematics is in its way, the poetry of logical ideas. —Albert Einstein

Problem From Heck # 95

Three circles of unit radius and centered at $\left ( 0,1.5 \right )$, $\left ( 0,-1.5 \right )$, and $\left ( 2,0 \right )$ are shown in dark blue color in the attached figure. The set of all circles that are tangent to all three of the given circles are drawn. What is the radius of the second largest such circle?

Inside a rectangular room, measuring 30 feet in length and 12 feet in width and height, a spider is at a point on the middle of one of the end walls, 1 foot from the ceiling, as at A; and a fly is on the opposite wall, 1 foot from the floor in the center, as shown at B. What is the shortest distance that the spider must crawl in order to reach the fly, which remains stationary? Of course the spider never drops or uses its web, but crawls freely.

Though this problem was much discussed in the Daily Mail from 18th January to 7th February 1905, when it appeared to create great public interest, it was actually first propounded by me in the Weekly Dispatch of 14th June 1903.

Solution:

Three circles of unit radius and centered at $\left ( 0,1.5 \right )$, $\left ( 0,-1.5 \right )$, and $\left ( 2,0 \right )$ are shown in dark blue color in the attached figure. The set of all circles that are tangent to all three of the given circles are drawn. What is the radius of the second largest such circle?